On the coordinate plane, let P $( t , \sin t ) ( 0 < t < \pi )$ be a point on the curve $y = \sin x$. Let circle $C$ be centered at P and tangent to the $x$-axis. Let Q be the point where circle $C$ is tangent to the $x$-axis, and let R be the point where circle $C$ meets segment OP.
If $\lim _ { t \rightarrow 0 + } \frac { \overline { \mathrm { OQ } } } { \overline { \mathrm { OR } } } = a + b \sqrt { 2 }$, find the value of $a + b$. (Here, O is the origin, and $a , b$ are integers.) [3 points]

What is the value of $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + 2 x - 8 } { x - 2 }$? [2 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10

For the function $f ( x ) = x ^ { 4 } + 3 x - 2$, what is the value of $f ^ { \prime } ( 2 )$? [3 points]
(1) 35
(2) 37
(3) 39
(4) 41
(5) 43

Two polynomial functions $f ( x )$ and $g ( x )$ satisfy $$\lim _ { x \rightarrow 0 } \frac { f ( x ) + g ( x ) } { x } = 3 , \quad \lim _ { x \rightarrow 0 } \frac { f ( x ) + 3 } { x g ( x ) } = 2$$ For the function $h ( x ) = f ( x ) g ( x )$, what is the value of $h ^ { \prime } ( 0 )$? [4 points]
(1) 27
(2) 30
(3) 33
(4) 36
(5) 39

For the function $f ( x ) = \frac { x ^ { 2 } - 2 x - 6 } { x - 1 }$, find the value of $f ^ { \prime } ( 0 )$.

For the function $f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + x - 1$, what is the value of $f ^ { \prime } ( 1 )$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

A polynomial function $f(x)$ satisfies $$f'(x) = 3x(x-2), \quad f(1) = 6$$ Find the value of $f(2)$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For a cubic function $f(x)$ with leading coefficient 1, let the function $g(x)$ be $$g(x) = f\left(e^{x}\right) + e^{x}$$ The tangent line to the curve $y = g(x)$ at the point $(0, g(0))$ is the $x$-axis, and the function $g(x)$ has an inverse function $h(x)$. What is the value of $h'(8)$? [3 points]
(1) $\frac{1}{36}$
(2) $\frac{1}{18}$
(3) $\frac{1}{12}$
(4) $\frac{1}{9}$
(5) $\frac{5}{36}$

15. The tangent line to the curve $y = e ^ { x }$ at the point $( 0,1 )$ is perpendicular to the tangent line to the curve $y = \frac { 1 } { x } ( x > 0 )$ at point P. The coordinates of P are $\_\_\_\_$

Let $f ( x ) = x ^ { 3 } + ( a - 1 ) x ^ { 2 } + a x$. If $f ( x )$ is an odd function, then the equation of the tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ is
A. $y = - 2 x$
B. $y = - x$
C. $y = 2 x$
D. $y = x$

The equation of the tangent line to the curve $y = 2 \ln ( x + 1 )$ at the point $( 0,0 )$ is $\_\_\_\_$.

The slope of the tangent line to the curve $y = ( ax + 1 ) e ^ { x }$ at the point $( 0,1 )$ is $- 2$. Then $a = $ $\_\_\_\_$.

13. The equation of the tangent line to the curve $y = 3 \left( x ^ { 2 } + x \right) \mathrm { e } ^ { x }$ at the point $( 0,0 )$ is $\_\_\_\_$.

Given the function $f ( x ) = \left( \frac { 1 } { x } + a \right) \ln ( 1 + x )$.
(1) When $a = - 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$.
(2) Do there exist $a$ and $b$ such that the curve $y = f \left( \frac { 1 } { x } \right)$ is symmetric about the line $x = b$? If they exist, find the values of $a$ and $b$. If they do not exist, explain why.
(3) If $f ( x )$ has an extremum on $( 0 , + \infty )$, find the range of $a$.

Given function $f ( x ) = \mathrm { e } ^ { x } - a x - a ^ { 3 }$.
(1) When $a = 1$, find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$;
(2) If $f ( x )$ has a local minimum value that is negative, find the range of values for $a$.

The function $h : x \mapsto x \cdot \ln \left( x ^ { 2 } \right)$ is given with maximum domain $D _ { h }$. (1a) [2 marks] State $D _ { h }$ and show that for the term of the derivative function $h ^ { \prime }$ of $h$ the following holds: $h ^ { \prime } ( x ) = \ln \left( x ^ { 2 } \right) + 2$.
(1b) [3 marks] Determine the coordinates of the local maximum point of the graph of $h$ located in the second quadrant.
Figure 1 shows the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ of a polynomial function $f$ defined on $\mathbb { R }$. Only at the points $\left( - 4 \mid f ^ { \prime } ( - 4 ) \right)$ and $\left( 5 \mid f ^ { \prime } ( 5 ) \right)$ does the graph $G _ { f ^ { \prime } }$ have horizontal tangents. [Figure]
(2a) [2 marks] Justify that $f$ has exactly one inflection point.
(2b) [2 marks] There are tangents to the graph of $f$ that are parallel to the angle bisector of the first and third quadrants. Using the graph $G _ { f ^ { \prime } }$ of the derivative function $f ^ { \prime }$ in Figure 1, determine approximate values for the $x$-coordinates of those points where the graph of $f$ has such a tangent.
The functions $f : x \mapsto x ^ { 2 } + 4$ and $g _ { m } : x \mapsto m \cdot x$ with $m \in \mathbb { R }$ are given, both defined on $\mathbb { R }$. The graph of $f$ is denoted by $G _ { f }$ and the graph of $g _ { m }$ by $G _ { m }$. (3a) [3 marks] Sketch $G _ { f }$ in a coordinate system. Calculate the coordinates of the common point of the graphs $G _ { f }$ and $G _ { 4 }$.
(3b) [2 marks] There are values of $m$ for which the graphs $G _ { f }$ and $G _ { m }$ have no common point. State these values of $m$.
The function $g$ is given by $g ( x ) = 0,7 \cdot e ^ { 0,5 x } - 0,7$ with $x \in \mathbb { R }$. The function $g$ is invertible. Figure 2 shows the graph $G _ { g }$ of $g$ and part of the graph $G _ { h }$ of the inverse function $h$ of $g$. [Figure]
(4a) [2 marks] Draw the missing part of $G _ { h }$ in Figure 2. Sub-task Part A 4b $( 2 \mathrm { marks } )$ Consider the region enclosed by the graphs $G _ { g }$ and $G _ { h }$. Shade the part of this region whose area can be calculated using the term $2 \cdot \int _ { 0 } ^ { 2,5 } ( x - g ( x ) ) \mathrm { dx }$.
Sub-task Part A 4c $( 2 \mathrm { marks } )$ State the term of an antiderivative of the function $k : x \mapsto x - g ( x )$ defined on $\mathbb { R }$.
The function $f : x \mapsto \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 1 }$ is given, defined on $\mathbb { R }$; Figure 1 (Part B) shows its graph $G _ { f }$. [Figure]

The function $f : x \mapsto 8 x ^ { 3 } + 3 x$ is defined on $\mathbb { R }$ with derivative function $f ^ { \prime }$. (1a) [2 marks] Calculate $f ^ { \prime } ( 1 )$.
(1b) [3 marks] Determine a term for the antiderivative $F$ of $f$ whose graph passes through the point ( $- 1 \mid 5$ ).
The figure shows the graph $G _ { g }$ of the function $g$ defined on $\mathbb { R }$ with $g ( x ) = 2 \cdot \sin \left( \frac { 1 } { 2 } x \right)$. [Figure]
(2a) [2 marks] Using the figure, assess whether the value of the integral $\int _ { - 2 } ^ { 8 } g ( x ) \mathrm { dx }$ is negative.
(2b) [3 marks] Prove by calculation that the following statement is true: The tangent to $G _ { g }$ at the origin is the line through the points $( - 1 \mid - 1 )$ and $( 1 \mid 1 )$.
Consider the family of functions $f _ { a }$ defined on $\mathbb { R }$ with $f _ { a } ( x ) = x \cdot e ^ { a x }$ and $a \in \mathbb { R } \backslash \{ 0 \}$. For each value of $a$, the function $f _ { a }$ has exactly one extremum.
(3a) [2 marks] Justify that the graph of $f _ { a }$ lies below the x-axis for $x < 0$. (3b) [3 marks] The displayed graphs I and II are graphs of the family; one of the two belongs to a positive value of $a$. Decide which graph this is and justify your decision. [Figure]
(4a) [2 marks] Give a term for a function $g$ defined on $\mathbb { R }$ that has range $[ - 2 ; 4 ]$.
(4b) [3 marks] Give a term for a function $h$ defined on $\mathbb { R }$ such that the term $\sqrt { h ( x ) }$ is defined exactly for $x \in [ - 2 ; 4 ]$. Explain the reasoning underlying your answer.
The figure shows a section of the graph $G$ of the function $f$ defined on $\mathbb { R } \backslash \{ - 3 \}$ with $f ( x ) = x - 3 + \frac { 5 } { x + 3 }$. $G$ has exactly one minimum point $T$. [Figure]

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$
We denote by $P_n^{(n)}$ the polynomial derived $n$ times of $P_n$.
Determine the degree of $P_n^{(n)}$ and calculate $P_n^{(n)}(1)$.

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Show that $\delta(f)$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}_+^*$. Compare $(\delta(f))'$ and $\delta(f')$.

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Explain why, for every $x > 0$, there exists a $y_1 \in \left]0, 1\right[$ such that $$(\delta(f))(x) = f'(x + y_1)$$

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Deduce that for every $x > 0$, for every $n \in \mathbb{N}^*$, there exists a $y_n \in \left]0, n\right[$ such that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ One may proceed by induction on $n \in \mathbb{N}^*$ and use the three preceding questions.

Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$. Show $$\forall x \in I, \quad 2f^{\prime}(x) = f(x)^2 + 1.$$

Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$, and set $\alpha_n = f^{(n)}(0) = P_n(0)$ for every natural integer $n$. Using the identity $2f^{\prime}(x) = f(x)^2 + 1$, show $2\alpha_1 = \alpha_0^2 + 1$ and $$\forall n \in \mathbb{N}^{\star}, \quad 2\alpha_{n+1} = \sum_{k=0}^{n} \binom{n}{k} \alpha_k \alpha_{n-k}.$$

Let $g$ be the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$ with radius of convergence $R \geqslant \pi/2$. Show $$\forall x \in I, \quad 2g^{\prime}(x) = g(x)^2 + 1.$$

Let $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$ and $g$ the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$. Both satisfy $2h^{\prime}(x) = h(x)^2 + 1$. By considering the functions $\arctan f$ and $\arctan g$, show $$\forall x \in I, \quad f(x) = g(x).$$